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Mean field electrodynamics (MFE) facilitates practical modeling of secular,large scale properties of astrophysical or laboratory systems withfluctuations.Practitioners commonly assume wide scale separation between meanand fluctuating quantities, to justify equality of ensemble and spatial ortemporal averages.Often however, real systems do not exhibit such scaleseparation. This raises two questions: (I) what are the appropriate generalizedequations of MFE in the presence of mesoscale fluctuations? (II) how preciseare theoretical predictions from MFE? We address both by first deriving theequations of MFE for different types of averaging, along with mesoscalecorrection terms that depend on the ratio of averaging scale to variation scaleof the mean. We then show that even if these terms are small, predictions ofMFE can still have a significant precision error. This error has an intrinsiccontribution from the dynamo input parameters and a filtering contribution fromdifferences in the way observations and theory are projected through themeasurement kernel.Minimizing the sum of these contributions can produce anoptimal scale of averaging that makes the theory maximally precise.Theprecision error is important to quantify when comparing to observations becauseit quantifies the resolution of predictive power. We exemplify these principlesfor galactic dynamos, comment on broader implications, and identifypossibilities for further work.

Derivation and precision of mean field electrodynamics with mesoscale fluctuations